Factor the following expression: $7$ $x^2+$ $32$ $x$ $-15$
Solution: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(7)}{(-15)} &=& -105 \\ {a} + {b} &=& & & {32} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-105$ and add them together. Remember, since $-105$ is negative, one of the factors must be negative. The factors that add up to ${32}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-3}$ and ${b}$ is ${35}$ $ \begin{eqnarray} {ab} &=& ({-3})({35}) &=& -105 \\ {a} + {b} &=& {-3} + {35} &=& 32 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {7}x^2 {-3}x +{35}x {-15} $ Group the terms so that there is a common factor in each group: $ ({7}x^2 {-3}x) + ({35}x {-15}) $ Factor out the common factors: $ x(7x - 3) + 5(7x - 3) $ Notice how $(7x - 3)$ has become a common factor. Factor this out to find the answer. $(7x - 3)(x + 5)$